\subsection{定积分定义}

	\begin{ti}
		$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{\sqrt{n^{2} + kn}} = $\htwo.
	\end{ti}

	\begin{ti}
		已知 $f(x) = a^{x^{3}}, a > 0$ 且 $a \ne 1$. 求
		\[
			\lim_{n \to \infty} \frac{1}{n^{4}}  \ln \left[ f(1) f(2) \cdots f(n) \right].
		\]
	\end{ti}

	\begin{ti}
		$\lim_{n \to \infty} \frac{\sqrt[n]{(n + 1) (n + 2) \cdots (n + n)}}{n} = $\htwo.
	\end{ti}

	\begin{ti}
		$f(x) = \begin{cases}
			\ee^{-x}, & x \ne 0,\\
			\lim_{n \to \infty} 2 \sum_{k=1}^{n} \frac{n}{(n + k)^{2}}, & x = 0,
		\end{cases}$ 求 $f'(0)$.
	\end{ti}

	\begin{ti}
		$\lim_{n \to \infty} \sin \frac{\uppi}{n} \sum_{k=1}^{n} \frac{1}{2 + \cos \frac{k \uppi}{n}} = $\htwo.
	\end{ti}

	\begin{ti}
		$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{n + \frac{(k - 1)^{2} + 1}{n}} = $\htwo.
	\end{ti}

	\begin{ti}
		$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{3^{\frac{k}{n}}}{n + \frac{1}{k}} = $\htwo.
	\end{ti}

	\begin{ti}
		设 $f(x) = \begin{cases}
			\lim_{n \to \infty} \sum_{k=1}^{n} \frac{|x|^{k/n}}{n + \frac{k}{n}}, & x \ne 0,\\
			0, & x = 0,
		\end{cases}$ 求 $f'(x)$.
	\end{ti}